Skipper Byrne headed home a free-kick from Grant McCann to sink the hopes of Tranmere, who seemed set for a play-off place themselves after leading by a 39th-minute goal from Craig Curran.
Curran headed home at the far post a cross from striker Ian Moore.
Only a super reaction save from goalkeeper Josh Lillis prevented Tranmere making it 2-0 on the stroke of half-time with an acrobatic shot from Steve Jennings.
Scunthorpe dominated the second half but could not find their finishing touch with leading marksman Garry Hooper missing a hat-trick of good chances.
But Scunthorpe finally struck with just two minutes remaining after Rovers had been reduced to 10 men when defender Gareth Edds had been sent off for a second bookable offence.